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<Title>Boost Graph Library: Push-Relabel Maximum Flow</Title>
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<H1><A NAME="sec:push_relabel_max_flow">
<TT>push_relabel_max_flow</TT>
</H1>

<P>
<PRE>
<i>// named parameter version</i>
template &lt;class Graph, class P, class T, class R&gt;
typename property_traits&lt;CapacityEdgeMap&gt;::value_type
push_relabel_max_flow(Graph&amp; g,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor src,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor sink,
   const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>)

<i>// non-named parameter version</i>
template &lt;class Graph,
	  class CapacityEdgeMap, class ResidualCapacityEdgeMap,
	  class ReverseEdgeMap, class VertexIndexMap&gt;
typename property_traits&lt;CapacityEdgeMap&gt;::value_type
push_relabel_max_flow(Graph&amp; g,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor src,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor sink,
   CapacityEdgeMap cap, ResidualCapacityEdgeMap res,
   ReverseEdgeMap rev, VertexIndexMap index_map)
</PRE>

<P>
The <tt>push_relabel_max_flow()</tt> function calculates the maximum flow
of a network. See Section <a
href="./graph_theory_review.html#sec:network-flow-algorithms">Network
Flow Algorithms</a> for a description of maximum flow.  The calculated
maximum flow will be the return value of the function. The function
also calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in
<i>E</i>, which are returned in the form of the residual capacity
<i>r(u,v) = c(u,v) - f(u,v)</i>.

<p>
There are several special requirements on the input graph and property
map parameters for this algorithm. First, the directed graph
<i>G=(V,E)</i> that represents the network must be augmented to
include the reverse edge for every edge in <i>E</i>.  That is, the
input graph should be <i>G<sub>in</sub> = (V,{E U
E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt>
must map each edge in the original graph to its reverse edge, that is
<i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>. The
<tt>CapacityEdgeMap</tt> argument <tt>cap</tt> must map each edge in
<i>E</i> to a positive number, and each edge in <i>E<sup>T</sup></i>
to 0.

<p>
This algorithm was developed by <a
href="./bibliography.html#goldberg85:_new_max_flow_algor">Goldberg</a>.


<H3>Complexity</H3>

The time complexity is <i>O(V<sup>3</sup>)</i>.


<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/push_relabel_max_flow.hpp"><TT>boost/graph/push_relabel_max_flow.hpp</TT></a>

<P>

<h3>Parameters</h3>

IN: <tt>VertexListGraph&amp; g</tt>
<blockquote>
  A directed graph. The
  graph's type must be a model of <a
  href="./VertexListGraph.html">Vertex List Graph</a>. For each edge
  <i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also
  be in the graph.
</blockquote>

IN: <tt>vertex_descriptor src</tt>
<blockquote>
  The source vertex for the flow network graph.
</blockquote>

IN: <tt>vertex_descriptor sink</tt>
<blockquote>
  The sink vertex for the flow network graph.
</blockquote>

<h3>Named Parameters</h3>

IN: <tt>capacity_map(EdgeCapacityMap cap)</tt>
<blockquote>
  The edge capacity property map. The type must be a model of a
  constant <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
  key type of the map must be the graph's edge descriptor type.<br>
  <b>Default:</b> <tt>get(edge_capacity, g)</tt>
</blockquote>

OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt>
<blockquote>
  The edge residual capacity property map. The type must be a model of
  a mutable <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
  key type of the map must be the graph's edge descriptor type.<br>
  <b>Default:</b> <tt>get(edge_residual_capacity, g)</tt>
</blockquote>

IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt>
<blockquote>
  An edge property map that maps every edge <i>(u,v)</i> in the graph
  to the reverse edge <i>(v,u)</i>. The map must be a model of
  constant <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
  key type of the map must be the graph's edge descriptor type.<br>
  <b>Default:</b> <tt>get(edge_reverse, g)</tt>
</blockquote>

IN: <tt>vertex_index_map(VertexIndexMap index_map)</tt>
<blockquote>
  Maps each vertex of the graph to a unique integer in the range
  <tt>[0, num_vertices(g))</tt>. The map must be a model of constant <a
  href="../../property_map/doc/LvaluePropertyMap.html">LvaluePropertyMap</a>. The
  key type of the map must be the graph's vertex descriptor type.<br>
  <b>Default:</b> <tt>get(vertex_index, g)</tt>
    Note: if you use this default, make sure your graph has
    an internal <tt>vertex_index</tt> property. For example,
    <tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
    not have an internal <tt>vertex_index</tt> property.
   <br>
</blockquote>


<h3>Example</h3>

This reads in an example maximum flow problem (a graph with edge
capacities) from a file in the DIMACS format. The source for this
example can be found in <a
href="../example/max_flow.cpp"><tt>example/max_flow.cpp</tt></a>.

<pre>
#include &lt;boost/config.hpp&gt;
#include &lt;iostream&gt;
#include &lt;string&gt;
#include &lt;boost/graph/push_relabel_max_flow.hpp&gt;
#include &lt;boost/graph/adjacency_list.hpp&gt;
#include &lt;boost/graph/read_dimacs.hpp&gt;

int
main()
{
  using namespace boost;

  typedef adjacency_list_traits&lt;vecS, vecS, directedS&gt; Traits;
  typedef adjacency_list&lt;vecS, vecS, directedS,
    property&lt;vertex_name_t, std::string&gt;,
    property&lt;edge_capacity_t, long,
      property&lt;edge_residual_capacity_t, long,
	property&lt;edge_reverse_t, Traits::edge_descriptor&gt; &gt; &gt;
  &gt; Graph;

  Graph g;
  long flow;

  property_map&lt;Graph, edge_capacity_t&gt;::type
    capacity = get(edge_capacity, g);
  property_map&lt;Graph, edge_reverse_t&gt;::type
    rev = get(edge_reverse, g);
  property_map&lt;Graph, edge_residual_capacity_t&gt;::type
    residual_capacity = get(edge_residual_capacity, g);

  Traits::vertex_descriptor s, t;
  read_dimacs_max_flow(g, capacity, rev, s, t);

  flow = push_relabel_max_flow(g, s, t);

  std::cout &lt;&lt; "c  The total flow:" &lt;&lt; std::endl;
  std::cout &lt;&lt; "s " &lt;&lt; flow &lt;&lt; std::endl &lt;&lt; std::endl;

  std::cout &lt;&lt; "c flow values:" &lt;&lt; std::endl;
  graph_traits&lt;Graph&gt;::vertex_iterator u_iter, u_end;
  graph_traits&lt;Graph&gt;::out_edge_iterator ei, e_end;
  for (boost::tie(u_iter, u_end) = vertices(g); u_iter != u_end; ++u_iter)
    for (boost::tie(ei, e_end) = out_edges(*u_iter, g); ei != e_end; ++ei)
      if (capacity[*ei] &gt; 0)
        std::cout &lt;&lt; "f " &lt;&lt; *u_iter &lt;&lt; " " &lt;&lt; target(*ei, g) &lt;&lt; " "
                  &lt;&lt; (capacity[*ei] - residual_capacity[*ei]) &lt;&lt; std::endl;
  return 0;
}
</pre>
The output is:
<pre>
c  The total flow:
s 4

c flow values:
f 0 1 4
f 1 2 4
f 2 3 2
f 2 4 2
f 3 1 0
f 3 6 2
f 4 5 3
f 5 6 0
f 5 7 3
f 6 4 1
f 6 7 1
</pre>

<h3>See Also</h3>

<a href="./edmonds_karp_max_flow.html"><tt>edmonds_karp_max_flow()</tt></a><br>
<a href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>.

<br>
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<TABLE>
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<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
</TD></TR></TABLE>

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